Infinite numbers are strange
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I watched a Youtube video recently and one idea from it has stuck in my head as very strange. I assume everyone's familiar with the idea that if you divide 1 by 7, you get an infinite decimal extending to the right... 1 / 7 = 0.14285714285714285714285714285714... But what if you take that the repeating 6 digit sequence indicated and repeat it infinitely to the right followed by a 3... ...2857142857142857142857142857143 That is clearly an infinity (it has infinitely many digits!), but if you multiply it by 7...
7*3 => 21 => 1 carry 2
7*4 => 28 + carried 2 => 30 => 0 carry 3
7*1 => 7 + carried 3 +> 10 => 0 carry 1
7*7 => 49 + carried 1 => 50 => 0 carry 5
7*5 => 35 + carried 5 => 40 => 0 carry 4
7*8 => 56 + carried 4 => 60 => 0 carry 6
7*2 => 14 + carried 6 => 20 => 0 carry 2
7*4 => 28 + carried 2 => 30 => 0 carry 3...Ultimately, you get... ...0000000000000000000000000000001 You have an infinite number of zeroes followed by 1, which is just 1. So this infinite number times 7 equals 1, which means it's also 1/7! I'd always been told that multiplying infinity by any number resulted in infinity, but this is clearly an infinite number which when multiplied by 7 is 1!
Obligatory SMBC[^]. :)
"These people looked deep within my soul and assigned me a number based on the order in which I joined." - Homer
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You cannot add a 3 to the end of an infinite number, as that converts it to a finite one.
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englebart wrote:
This is why there are fixed decimals that always round in favor of the bank.
This is incorrect. Bank accounts use "round to nearest or away", where fractional cents are rounded to the nearest value (up or down). If the residue is exactly 0.5 cents, the number is rounded "away" - up for positive, down for negative. If you are running a credit, this gives you a tiny statistical advantage. If you are running a debit, this gives the bank a tiny statistical advantage. In neither case is this likely to have a measurable effect, unless you aggregate over billions of operations a day.
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows. -- 6079 Smith W.
Daniel Pfeffer wrote:
Bank accounts use "round to nearest or away", where fractional cents are rounded to the nearest value (up or down)
I couldn't find any regulation that specifies how banks round numbers. And given 'banks' exist throughout the world I suspect certainly in some places rounding might use different rules. I worked for a financial company (not a bank) and the rounding was decided by me. There are multiple rules. Last time I looked (and can recall) there are three different types of rounding suited to financial transactions. Rather than scientific. Although one of those might also be scientific. Two of them provide 'better' results than just rounding up on '0.5' otherwise down. Which I suspect you are referring to. That specific method tends to favor a specific result. Because there are 6 digits from 5-9 but only 4 from 0-4.
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Chris Maunder wrote:
There are actually many different "infinities". The numbers 1,2,3...is an infinite set. The set of real numbers between 1 and 2 (eg 1.1, 1.01, 1.001 and on and on) is also infinite, and large than the set of integers. One infinity can be bigger than another infinity. Even though they are both infinite.
Actually, I've never been totally convinced of this, although I'm open to it being proved in some way. The only way I've ever seen is Cantor's Diagonalization, which says to take a list of all the Real (Rational plus Transcendental, etc) numbers between zero and one then to create a new number by taking the first decimal digit of the first number, second decimal digit of the second number, third of the third, etc. The argument is that this number cannot be on the list, so therefore you can put the infinite number of Reals into correspondence with the Integers so there there is at least a Countable infinity (number of integers) and an Uncountable infinity (number of Reals). My scepticism comes from the statement about creating the list of Reals. I'd like to use the following pseudo code:
reals = New List
reals.Add(0.1)
reals.Add(0.5)
reals.Add(pi)
// as many as you wantrepeat forever // until list is complete
for each number r in reals
x = new real
for each decimal digit p of r
digit p of x = not_the_same_as(digit q of r) // function elsewhere
if x not in reals
reals.Add(x)
else
terminate // countable list of reals is completeMy point is that this procedure *is* Cantor's Diagonalization so if that can find another Real to add, then the list building shouldn't have terminated and, if it can't, then it's not been proved that there are more Reals than Integers. It may be true, but this doesn't prove it. I feel sure there must be an alternative proof to Cantor's, but I've never found it. Perhaps it relies on maths I've never encountered and would need a degree in Mathematics to understand! (Mine was Physics! :) ) FYI, I recall I once saw a proof that there are more Transcendental numbers (like pi or e) than Rational numbers (like 1/5, 3/7, etc) but can't recall if it was also based on Cantor's method.
StarNamer@work wrote:
It may be true, but this doesn't prove it.
Interesting argument. However it fails from the concept of terms versus proof. An infinite series is just a concept. Since of course you can't reach the end. As you are suggesting. So one must accept the concept without enumerating the set. But once one accepts the concept then one can discuss it. Thus there is a set that does have all of the numbers (but still conceptually infinite) and thus there can be a comparative enumeration using the other set.
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StarNamer@work wrote:
Many years ago, I researched for a PhD in Nuclear Structure Physics and studied some High-Energy (particle) Physics so am aware of renormalization to get rid of infinites in theories, the meaning (or lack of it) of anything divided by zero, etc.
As you are no doubt aware, there are no infinities in nature. Any infinity that does turn up in our theories is a sign that the theory is incomplete. Renormalization is a clever mathematical trick that allows us to make predictions without having a complete theory, but one must remember its limitations - it is a mathematical trick, not part of a correct description of nature.
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows. -- 6079 Smith W.
Daniel Pfeffer wrote:
As you are no doubt aware, there are no infinities in nature
Not sure I agree with that. Time exists because the universe exists. If the universe ends then time ends. If the universe never ends then time never ends. It becomes infinite. Your statement above is an affirmative belief (not a proof) that the universe must end.
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Daniel Pfeffer wrote:
Bank accounts use "round to nearest or away", where fractional cents are rounded to the nearest value (up or down)
I couldn't find any regulation that specifies how banks round numbers. And given 'banks' exist throughout the world I suspect certainly in some places rounding might use different rules. I worked for a financial company (not a bank) and the rounding was decided by me. There are multiple rules. Last time I looked (and can recall) there are three different types of rounding suited to financial transactions. Rather than scientific. Although one of those might also be scientific. Two of them provide 'better' results than just rounding up on '0.5' otherwise down. Which I suspect you are referring to. That specific method tends to favor a specific result. Because there are 6 digits from 5-9 but only 4 from 0-4.
Here you are: [Converting to the euro](https://economy-finance.ec.europa.eu/euro/enlargement-euro-area/adoption-fixed-euro-conversion-rate/converting-euro\_en) When the EU introduced the Euro, conversion from the national currencies to the Euro was done using to the rule that I mentioned (perform the calculation exactly, then round to 2 decimal places using 'round to nearest or away'). I assumed that this was the rule used by most banking operations. Note that the IEEE Std 754-2019 Floating-Point Standard specifies decimal, as well as binary, floating-point. Parts of the decimal specification (e.g. the "quantum" concept, and 'round to even or away') were added specifically in order to ease the decimal calculations performed by banks.
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows. -- 6079 Smith W.
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Daniel Pfeffer wrote:
As you are no doubt aware, there are no infinities in nature
Not sure I agree with that. Time exists because the universe exists. If the universe ends then time ends. If the universe never ends then time never ends. It becomes infinite. Your statement above is an affirmative belief (not a proof) that the universe must end.
There are only two possibilities: (a) the Universe has always existed, or (b) the Universe has a finite age. If the Universe has always existed, its age is infinite. Adding any finite amount to that value will still give infinity. If the Universe began at a finite point in the past, its age is finite. Adding any finite amount, however large, to that finite number will still yield a finite number. If our theories are correct, the Universe is approximately 14 billion years old. Adding a googolplex (1010100 years) to that will yield a very large, but still finite, number. Note that even the most far-out theories cannot say anything about how the Universe will look at that advanced age; the extrapolation from our current observations is simply too extreme.
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows. -- 6079 Smith W.
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There are only two possibilities: (a) the Universe has always existed, or (b) the Universe has a finite age. If the Universe has always existed, its age is infinite. Adding any finite amount to that value will still give infinity. If the Universe began at a finite point in the past, its age is finite. Adding any finite amount, however large, to that finite number will still yield a finite number. If our theories are correct, the Universe is approximately 14 billion years old. Adding a googolplex (1010100 years) to that will yield a very large, but still finite, number. Note that even the most far-out theories cannot say anything about how the Universe will look at that advanced age; the extrapolation from our current observations is simply too extreme.
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows. -- 6079 Smith W.
Daniel Pfeffer wrote:
There are only two possibilities: (a) the Universe has always existed, or (b) the Universe has a finite age.
I do not agree. That is a binary statement that is not necessarily true. The universe could have 'started' in the sense of the word meaning that before that point time did not exist and after that point time did.
Daniel Pfeffer wrote:
If the Universe has always existed, its age is infinite.
Yet you said "there are no infinities in nature." Are you claiming that time is not part of nature?
Daniel Pfeffer wrote:
If the Universe began at a finite point in the past, its age is finite
I don't agree with that at all. There are an infinite number of positive integers. Positive integers, specifically and by definition, start at zero. But they never end. The universe itself can only 'begin' when time starts. That is the nature of the word 'begin'. But the fact that it started does not mean that it must end.
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Here you are: [Converting to the euro](https://economy-finance.ec.europa.eu/euro/enlargement-euro-area/adoption-fixed-euro-conversion-rate/converting-euro\_en) When the EU introduced the Euro, conversion from the national currencies to the Euro was done using to the rule that I mentioned (perform the calculation exactly, then round to 2 decimal places using 'round to nearest or away'). I assumed that this was the rule used by most banking operations. Note that the IEEE Std 754-2019 Floating-Point Standard specifies decimal, as well as binary, floating-point. Parts of the decimal specification (e.g. the "quantum" concept, and 'round to even or away') were added specifically in order to ease the decimal calculations performed by banks.
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows. -- 6079 Smith W.
Daniel Pfeffer wrote:
Here you are
Which supports exactly what I said. Are there ways to round currencies - yes. Is there more than one way to do it - yes. I stated both of those. How does one decide which to use? By law, regulation or just by picking one. You provided an example where the method was specified for the very specific case. It specifically supports the different choices also. The following is a way to round currencies: "it is prohibited to round or truncate the conversion rate." Instead it provides the rule to be used: "if the number in the third decimal place is less than 5..." But it also recognizes other possibilities with the following: "Introduction of the euro may not alter the terms of legal instruments, " And this: "National law can bring more detail to rules on rounding as long as this leads to a higher degree of accuracy."
Daniel Pfeffer wrote:
Note that the IEEE Std 754-2019
You are merely pointing out ways that one can do it. Which I already said exist.
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I missed out all the boilerplate phrases along the lines of "As you increase the number of digits 7*0.14285714... tends towards 1 so, in the limit, is assumed to be 1...", etc.
Kornfeld Eliyahu Peter wrote:
0.1428573 * 7 = 1.0000011 0.14285731428573 * 7 = 1.00000120000011 --- 0.142857 * 7 = 0.999999 0.142857142857 * 7 = 0.999999999999
You've misread the first number (second in my message). It's... ........2857142857142857142857142857143 or ........2857142857142857142857142857142857142857142857142857142857143 That is, an integer with an infinite number of repetitions of ...285714... followed by 3. Although there's a notation for recurring decimals, I don't know of a shorthand for a p-adic number (which is what this is). The point is that I (and the video) didn't suggest stopping the calculation at any point. Obviously, if you do they aren't the same and, in the case of the 'infinite' integer, you don't actually have a result! It's only if you project to the theoretical limit that the results are equivalent. Many years ago, I researched for a PhD in Nuclear Structure Physics and studied some High-Energy (particle) Physics so am aware of renormalization to get rid of infinites in theories, the meaning (or lack of it) of anything divided by zero, etc. I'd just never encountered p-adic numbers[^] before I watched that video[^].
Finally I had time to watch the video... Now I understand you better... However I have to say that, that video cuts corners in a very horrible way IMHO... It seems that its simplifications and inclusions are chosen to server a specific end result, but not clear and whole or precise...
"If builders built buildings the way programmers wrote programs, then the first woodpecker that came along would destroy civilization." ― Gerald Weinberg
